1 Introduction

Rational polynomial coefficients (rational polynomial coefficients, RPC), in essence, is a rational function model (Rational Function Model-RFM). It can establish the relationship between image points and space coordinates, does not need internal and external orientation elements, avoids the geometric process of imaging, and can be widely used in linear array image processing. RFM expresses the coordinates of image points as the ratio of polynomials whose independent variables are the spatial coordinates of corresponding ground points.

The attitude control of the satellite during satellite imaging leads to the extremely complex form of the strict geometric model of the image (the so-called strict geometric model refers to the strict geometric model based on the traditional collinear equation). At the same time, the satellite's detailed orbital ephemeris, sensor imaging parameters, imaging methods and other information are delivered together, and the end user needs to have professional knowledge of photogrammetry and a complex application processing system. In order to reduce the demand for the user's professional level, expand the scope of users, and protect the core technical parameters of the satellite from being leaked, the RPC positioning model came into being.

RPC is a sensor-independent general-purpose imaging geometry model. RPC is a fitting form of the strict geometric model of the sensor. The strict geometric model here refers to the geometric model of the image-ground relationship constructed by the platform’s trajectory parameters, attitude parameters, sensor installation parameters and sensor internal geometric parameters measured by the platform load . Due to the inevitable errors of different natures in these parameters, the fitting model RPC also has corresponding errors. The traditional method of correcting RPC errors is to perform a polynomial correction on the image points projected from the ground point to the image space through RPC, so that the coordinates of the projected image points coincide with the coordinates of the measured image points, so as to eliminate the error.

2. RPC rational function model

The RPC model uses the ratio of polynomials to establish the relationship between the detector image point coordinates d(line, sample) (which can be understood as row and column numbers) and its corresponding ground imaging point coordinates D(latitude, lontitude, height). RPC is also a general expression form of various sensor geometric models. The positive form of RPC is
$\begin{aligned} & F_1=L_n=\frac{N um L(U, V, W)}{\operatorname{Den } L(U, V, W)} \\ & F_2=S_n=\frac{N um S(U, V, W)}{\operatorname{DenS}(U, V, W)} \end{aligned}$
式中: $\\ \quad \operatorname{Num} L(U, V, W)=a_1+a_2 V+a_3 U+a_4 W+a_5 V U+a_6 V W+a_7 U W+a_8 V^2+a_9 U^2+a_{10} W^2+a_{11} U V W+a_{12} V^3+a_{13} V U^2+a_{14} V W^2+a_{15} V^2 U+a_{16} U^3+a_{17} U W^2+a_{18} V^2 W+a_{19} U^2 W+a_{20} W^3 ;$
$\begin{gathered} \operatorname{Den} L(U, V, W)=b_1+b_2 V+, \cdots,+b_{19} U^2 W+b_{20} W^3 \\ \operatorname{Num} S(U, V, W)=c_1+c_2 V+, \cdots,+c_{19} U^2 W+c_{20} W^3 \\ \operatorname{DenS}(U, V, W)=d_1+d_2 V+, \cdots,+d_{19} U^2 W+d_{20} W^3 \end{gathered}$

在式中, $a_1, a_2, \cdots, a_{20}, b_1, b_2, \cdots, b_{19}, b_{20},c_1, c_2, \cdots, c_{19}, c_{20},d_1, d_2, \cdots, d_{19}, d_{20}$ , is the calculation coefficient. The left side of the equation is the image square coordinates: $L_n$ For the normalized row coordinates, let the row coordinates be $r$ , the line offset parameter is $LINE\_OFF$ , the line scaling parameter is LINE_SCALE.
$L_n=(r-{LINE\_OFF})/LINE\_SCALE;$

$S_n$ For the normalized column coordinates, let the column coordinates be $c$ , the column offset parameter is SAMP_OFF, and the column scaling parameter is SAMP_SCALE, then
$S_n=(c-{SAMP_{-}OFF})/SAMP_{-}SCALE ;$
the right side of equation (1)-(5) is the coordinate of object space: $U$ is the normalized latitude coordinate, let the latitude coordinate be $B$ , the latitude offset parameter is $LAT_{-}$ $OFF$ , the latitude scaling parameter is $LAT_{-}SCALE$ ,则
$U=(B{-LAT\_OFF})/LAT\_{}SCALE\text{;}$
$V$ is the normalized longitude coordinate, let the longitude coordinate be $L$ , the longitude offset parameter is $LONG_{-}OFF$ , the longitude scaling parameter is $LONG_{-}SCALE$ ,则
$V=\left(L-L{ONG}\_OFF\right)/LONG_{-}SCALE\text{;}$
$W$ is the normalized elevation coordinate, let the elevation coordinate be $H$ ,高程 $HEIGHT\_SCALE$ ,则
$W=\left(H-HEIGHT_{-}OFF\right)/HEIGHT_{-}SCALE$

Liu Jiang, Yue Qingxing, and Qiu Zhenge. "Research on RPC Correction Method." Remote Sensing for Land and Resources 1(2013):5.

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http://geotiff.maptools.org/rpc_prop.html

3. Excellent RPC-related knowledge blog posts

3.1 Information about RPC and RPB files

RPC and RPB are only different in format, but the content and basic parameters are the same.

RPC file naming format: filename_rpc.txt or filename_RPC.txt
RPB file naming format: filename.rpb

Blog post: Remote sensing RPC, RPB file related information gives a detailed introduction and samples.

3.2 Matlab realizes RPC forward calculation and reverse calculation

Blog post: Matlab realizes RPC forward and reverse calculation provides Matlab code. The function is as follows:

Main function (pixel -> latitude and longitude height)
RPC positive function (latitude and longitude height -> pixel coordinates)
RPC inverse calculation function (pixel -> latitude and longitude):

3.3 Batch image RPC orthorectification using Gdal

Blog post: (Python) Batch image RPC orthorectification using Gdal introduces the use of Python to perform orthorectification of the RPC model, and finally achieves the effect of batch orthorectification.

3.4 (Python) satellite RPC rational polynomial model reading and forward and backward projection coordinate calculation principle and implementation

Blog post: (Python) Satellite RPC Rational Polynomial Model Reading and Forward and Backward Projection Coordinate Calculation Principle and Implementation Introduces the basic knowledge of the RPC geometric positioning model, then provides the implementation of the RPC model code and conducts a simple demonstration of its use, and finally evaluates it The code's precision versus performance.

3.5 The third round of high-score image batch processing - RPC file and geometric correction

Blog post: The third batch of high-score image batch processing - RPC file and geometric correction

4. Reference

https://blog.csdn.net/stone_tigerLI/article/details/122123424

RPC model